Unit 5 Logarithmic Exponential and Other Transcendental Functions

Unit 5 Logarithmic, Exponential and Other Transcendental Functions Review Problems Calculus 5 -R

Find the domain of the function: f(x) = ln(3 x + 1). Find the domain of the function: f(x) = 3 + ln(x - 1). (1, � ) 1 Review Problems

Match the graph with the correct function [A] f(x) = ln x [B] f(x) = ex-1 [C] f(x) = ln(x - 1) [D] f(x) = ex 2 Review Problems

Sketch the graph: f(x) = ln|x|. Solve for x: ln(5 x + 1) + ln x = ln 4 3 Review Problems

Solve for x: ln(5 x - 1) - ln x = 3. dy/dx for y = ln(5 - x)6 4 Review Problems

Find the derivative: f(x) = ln(x 3 + 3 x)3 Find the derivative: 5 Review Problems

Find the derivative: 6 Review Problems

Differentiate: y = ln(ln tan x) Find y’ y = ln|2 x 2 - 5| 7 Review Problems

Find y’ if ln xy = x + y Use logarithmic differentiation to find 8 Review Problems

Find the slope of the tangent line to the graph of y = ln x 2 at the point where x = e 2 Evaluate the integral: ln 4 9 Review Problems

Evaluate the integral: ln|ax + b| + C Evaluate the integral: -2 10 Review Problems

Evaluate the integral: x+ 11 ln(x 2 + 1) + C Review Problems

Evaluate the integral: 8 x + ln(x 2 + 1) + C Evaluate the integral: 9 x - 12 ln(x 2 + 1) + C Review Problems

Evaluate the integral: +C Evaluate the integral: ln|sec 3 x| + C 13 Review Problems

Evaluate the integral: -2 cos x + ln|csc x + cot x| + C Evaluate the integral: ln|tan x| + C 14 Review Problems

Evaluate the integral: Differentiate: 15 Review Problems

Match the graph shown with the correct function [A] f(x) = e (x-1) [B] f(x) = e-(x-1) [C] f(x) = ex + 1 [D] f(x) = e-x + 1 16 Review Problems

Differentiate: 17 Review Problems

Find: if xey + 1 = xy Evaluate the integral: +C 18 Review Problems

Find the slope of the tangent line to the graph of y = (ln x)ex at the point where x = 2 Evaluate the integral: -ecosx + C 19 Review Problems

Evaluate the integral: -95 e-t/5 + C Evaluate the integral: +C 20 Review Problems

Find if y = 3 xx 3 3 xx 2[3 + (ln 3)x] Differentiate: y = x 1 -x 21 Review Problems

Differentiate y = xx xx[1 + ln x] Evaluate the integral: +C 22 Review Problems

Find the area bounded by the function f(x) = 2 -x, the x-axis, x = -2, and x = 1 A certain type of bacteria increases continuously at a rate proportional to the number present. If there are 500 present at a given time and 1000 present 2 hours later, how many will there be 5 hours from the initial time given? 2828 23 Review Problems

A certain type of bacteria increases continuously at a rate proportional to the number present. If there are 500 present at a given time and 1000 present 2 hours later, how many hours (from the initial given time) will it take for the numbers to be 2500? Round your answer to 2 decimal places. 4. 64 A mold culture doubles its mass every three days. Find the growth model for a plate seeded with 1. 6 grams of mold. [Hint: Use the model y = Cekt where t is time in days and y is grams of mold. ] 1. 6 e 0. 23105 t 24 Review Problems

The balance in an account triples in 21 years. Assuming that interest is compounded continuously, what is the annual percentage rate? 5. 23% The balance in an account triples in 20 years. Assuming that interest is compounded continuously, what is the annual percentage rate? 5. 49% 25 Review Problems

A radioactive element has a half-life of 50 days. What percentage of the original sample is left after 85 days? 30. 78% A radioactive element has a half-life of 40 days. What percentage of the original sample is left after 48 days? 43. 53% 26 Review Problems

The number of fruit flies increases according to the law of exponential growth. If initially there are 10 fruit flies and after 6 hours there are 24, find the number of fruit flies after t hours. y = 10 eln(12/5)t/6 Determine whether the function y = 2 cos x is a solution to the differential equation No 27 Review Problems

verify that equation is a solution to the differential Find the particular solution to the differential equation given the general solution and the initial condition y = 1 - cos x 28 Review Problems

Find the particular solution to the differential equation given the general solution and the initial condition y(0) = 5. Use integration to find a general solution to the differential equation y = (x + 1)3/2 29 Review Problems +C

Use integration to find a general solution to the differential equation y = 3 ln|1 + x| + C Use integration to find a general solution to the differential equation y= 30 Review Problems

Find the general solution to the first-order differential equation: (4 - x)dy + 2 y dx = 0 y = C(4 - x)2 Find the general solution to the first-order differential equation: x cos 2 y + tan y =0 x 2 + sec 2 y = C 31 Review Problems

Find the general solution to the first-order differential equation: y dx + (y - x)dy = 0 y ln|y| + x = Cy Find the general solution to the first-order differential equation: y= 32 Review Problems

Find the general solution to the first-order differential equation: Find the particular solution of the differential equation that satisfies the initial condition y(0) = 7 y = 500 - 493 e-x 33 Review Problems

Find the solution to the initial value problem y(-1) = 0 ey + sin y = (x 2 + 1) Find the solution to the initial value problem y(1) = 0 ln(1 + y 2) = 2 ln x + x 2 - 1 34 Review Problems

A deposit of $1000 is made into a fund with an annual interest rate of 5 percent. Find the time (in years) necessary for the investment to double if the interest is compounded continuously. Round your answer to 2 decimal places. 13. 86 years A deposit of $1000 is made into a fund with an annual interest rate of 5 percent. Find the time (in years) necessary for the investment to triple if the interest is compounded continuously. Round your answer to 2 decimal places. 22. 24 years 35 Review Problems

Find the constant k so that the exponential function y = 3 ekt passes through the points given on the graph. Solve the differential equation: 36 Review Problems

Find the particular solution to the differential equation given the general solution and the initial condition y = 1 - cos x Find the general solution of the differential equation sin y + cos x = C 37 Review Problems

Find the general solution of the differential equation Evaluate: arccos 38 Review Problems

Find the exact value: 39 Review Problems

Find the exact value: sin(arctan 3) 40 Review Problems

Find the exact value: 41 Review Problems

Write an algebraic expression for tan[arcsin x]. Differentiate: 42 Review Problems

Differentiate: y = arctan ex 43 Review Problems

Differentiate: h(t) = arccos t 2 Differentiate: 44 Review Problems

Evaluate the integral: +C Evaluate the integral: arcsec|2 x| + C 45 Review Problems

Evaluate the integral: ln(x 2 + 9) + arctan +C Evaluate the integral: +C 46 Review Problems

Evaluate the integral: +C Evaluate the integral: arctan 47 Review Problems +C

Evaluate the integral: arctan +C Evaluate the integral: +C 48 Review Problems

Evaluate the integral: arcsin(x - 2) + C Evaluate the integral: +C 49 Review Problems

Evaluate the integral: +C Evaluate the integral: arcsec 50 Review Problems +C

Solve the differential equation: y= +C Solve the differential equation: y = Cearcsec x 51 Review Problems

1. (1, � ) 8. 2. C 9. 3. Graph 10. 4. 11. -2 5. 12. 8 x + 6. 13. 7. 14. ln 4 ln|ax + b| + C x+ ln(x 2 + 1) + C 9 x - +C ln(x 2 + 1) + C ln|sec 3 x| + C -2 cos x + ln|csc x + cot x| + C ln|tan x| + C Answers

xx[1 + ln x] 15. 22. 16. D 23. 2828 17. 24. 4. 64 1. 6 e 0. 23105 t 18. +C 19. +C 25. 5. 23% -ecosx + C 26. 30. 78% 20. -95 e-t/5 + C 21. 3 xx 2[3 + (ln 3)x] +C 5. 49% 43. 53% 27. y = 10 eln(12/5)t/6 28. y = 1 - cos x x 1 -x Answers No

29. 36. y = (x + 1)3/2 30. y = 3 ln|1 + x| + C 31. +C y= y = C(4 - x)2 x 2 + sec 2 y 37. 38. =C 32. y ln|y| + x = Cy y= 39. 33. 40. 34. y = 500 - 493 e-x ey + sin y = ln(1 + (x 2 + 1) y 2 ) = 2 ln x + 35. 13. 86 years 22. 24 years x 2 y = 1 - cos x -1 41. 42. Answers sin y + cos x = C

43. 50. 44. 51. y = +C 45. 46. arcsec|2 x| + C ln(x 2 + 9) + arctan 47. 48. arctan +C +C 49. arcsin(x - 2) + C 52. 53. 54. 55. 56. +C Answers +C +C arcsec +C y = Cearcsec x